Cotranscriptionally encoded RNA strand displacement circuits

Engineered molecular circuits that process information in biological systems could address emerging human health and biomanufacturing needs. However, such circuits can be difficult to rationally design and scale. DNA-based strand displacement reactions have demonstrated the largest and most computationally powerful molecular circuits to date but are limited in biological systems due to the difficulty in genetically encoding components. Here, we develop scalable cotranscriptionally encoded RNA strand displacement (ctRSD) circuits that are rationally programmed via base pairing interactions. ctRSD circuits address the limitations of DNA-based strand displacement circuits by isothermally producing circuit components via transcription. We demonstrate circuit programmability in vitro by implementing logic and amplification elements, as well as multilayer cascades. Furthermore, we show that circuit kinetics are accurately predicted by a simple model of coupled transcription and strand displacement, enabling model-driven design. We envision ctRSD circuits will enable the rational design of powerful molecular circuits that operate in biological systems, including living cells.


II. ctRSD gate design considerations
This section details the different design considerations analyzed during development of the ctRSD gates and describes the motivation for specific design choices. Section IIA compares two methods to transcriptionally encode RNA strand displacement gates: transcription of the two gate strands from separate transcription templates or transcription of an RNA hairpin with a ribozyme that cleaves the hairpin after folding to produce a dsRNA gate. The former method introduces a significant downstream leak reaction (section IIA) and was not used. Section IIB analyzes four different transcription paths for producing ctRSD gates. In principle, these different transcription paths are conceptually equivalent but depend on the selected toehold directionality (5′ vs 3′) and the position of the ribozyme within the transcript. Section IIC analyzes three different selfcleaving ribozyme options for the ctRSD gates. The desired reaction network. I1 releases an output from the 1_2r gate that in turn reacts with a DNA reporter complex to produce a fluorescent signal. The 1_2r gate should not react with the DNA reporter on its own. (B) The typical method for preparing DNA gates for TMSD circuits. The 1_2r gate and the DNA reporter are prepared in separate test tubes and then mixed to make the circuit. Separate preparation ensures the two complexes are kinetically precluded from reacting in the absence of input. (C) A method for preparing ctRSD gates in which separate transcription templates produce the output and gate′ strands of the gate. Here, the two RNA strands should hybridize after transcription to form the gate. However, if this is done alongside the DNA reporter, or any other downstream gates that take O2r as their input, the output strand can also react downstream before it hybridizes to form the gate. This introduces significant leak in the circuit ( fig. S8). (D) A method for preparing ctRSD gates in which a gate is encoded as a hairpin that cleaves into a dsRNA product via an internal self-cleaving ribozyme. In this method, the hairpin rapidly folds during transcription to prevent any downstream reaction in the absence of input. This method for preparing ctRSD gates was used in this study.  Increasing the gate′ strand template concentration results in excess gate′ transcript. The O2r strand did not stain well. Transcription and native gel electrophoresis were conducted as described in the Methods of the main text. After transcription, samples were incubated with DNase I for 1 h before gel electrophoresis.

B. The cotranscriptional folding pathway
Considering the directionality of the single-stranded RNA (ssRNA) toehold that facilitates strand displacement and the placement of the self-cleaving ribozyme within the RNA transcript, there are four possible designs for ctRSD gates ( fig. S10). Previous work indicates that 5′ toeholds on RNA strand displacement gates perform better than 3′ toeholds (35,36), so we focused on designs with 5′ toeholds (fig. S10, A and B). The placement of the self-cleaving ribozyme influences which domains of the gate are transcribed first. For example, placing the ribozyme adjacent to the 5′ toehold results in transcription of the output region (2-, b-, and 1-domains) of the gate first, while placing the ribozyme on the opposite side of the transcript results in transcription of the gate′ strand first. In many DNA strand displacement circuits the output sequences of the DNA gates are constrained to a 3 letter code (C, A, or, T). This constraint reduces the possibility of unwanted secondary structure from forming and preventing output strands in larger circuits from hybridizing with each other (crosstalk) (4,5). We adopted the same sequence constraints in our RNA gate designs, limiting the gate output sequences (2-, b-, and 1-domains) to only C, A, or U bases. This constraint is particularly important for RNA circuits because G-U wobble base pairings are more energetically favored in RNA than G-T wobble pairings in DNA (53). Thus, even output sequences constrained to G, A, U bases could fold into undesired secondary structures. In the ctRSD gate design in which the self-cleaving ribozyme is placed opposite of the gate toehold ( fig. S10B), the gate′ strand (a′-, 1′-, and b′domains), whose sequence would be composed of G, A, U bases, would be transcribed first.
Given that cotranscriptional folding of RNA is much faster than transcription (28)

C. Self-cleaving ribozyme selection
Three well characterized ribozymes were considered: the hammerhead ribozyme, the hairpin ribozyme, and the hepatitis delta virus (HDV) ribozyme. The HDV ribozyme has several advantages over the hammerhead and hairpin ribozymes. First, the HDV ribozyme folds quickly into a stable structure (40,49), likely making it resistant to misfolding across different flanking sequences. Second, the rate constant for HDV ribozyme cleavage has been reported as 52 min -1 in certain settings (41), compared to 1 min -1 for the hammerhead (54) or 0.5 min -1 to 0.05 min -1 for the hairpin ribozymes (54,55). Lastly, the HDV ribozyme has little sequence preference upstream of the cleavage site (39). Both the hammerhead (56) and hairpin (55) ribozymes have cleavage site sequence constraints and their cleavage sites are flanked by RNA duplexes thus requiring a dissociation step following cleavage to separate the two strands. This dissociation step is particularly problematic in our ctRSD gate designs, in which the ssRNA toehold for strand displacement must be exposed after cleavage. In our designs, the hammerhead and hairpin ribozymes require 6 and 4 bases, respectively, to dissociate after cleavage to expose the toehold for strand displacement ( fig. S11). In the case of the hammerhead ribozyme, these 6 bases are likely to remain hybridized most of the time after cleavage, impeding RNA strand displacement. The HDV ribozyme does not suffer from these sequence limitations, driving this choice for our designs. We found the HDV ribozyme (sequence adopted from Ref (39)) resulted in the desired efficient and rapid cleavage in our RNA gates (Figure 1 and fig. S12). We also tested a ctRSD gate with the hairpin ribozyme, but much less cleavage was observed than with the HDV ribozyme ( fig. S13). In the final cleavage products, the number of bases that must dissociate to expose the toehold are written in gold lettering. The HDV ribozyme does not require any bases to unhybridized after cleavage, making it the best candidate for our ctRSD gate design.  In vitro transcription reactions were conducted for 1 h, followed by DNase I digestion (Methods). Samples were mixed with an equal volume of Gel Loading Buffer II (Invitrogen), heated to 80 o C for 5 min, and subsequently run on 4 % EX agarose E-gels for 30 min. The 1_2 gate xRz and 1_2 HDV cau controls, which represent the full length uncleaved and cleaved products, respectively, are depicted in fig. S2. All samples were analyzed on the same gel but lanes between the HDV Rz and HP Rz samples were removed because they were not related to this experiment.

III.
Equilibrium analysis with NUPACK NUPACK 3.2.2 (42) was used for equilibrium analysis of RNA complexes. We used the default NUPACK parameters for RNA (1.0 mol/L Na+ and 0 mol/L Mg++, dangles: some). Although there is 6 mmol/L MgCl2 in our transcription buffer, there is a total of 8 mmol/L NTPs, which will sequester MgCl2, so the concentration of free Mg++ is unknown. For RNA analysis, the default salt conditions are the only options. Unless otherwise state, analysis was conducted at 37 o C with 1 µmol/L of each RNA species. Changing the equimolar concentration of the RNA species between 10 nmol/L and 100 µmol/L does not change the predicted equilibrium concentrations.
For analysis of the reaction I1 + 1_2 gate ↔ I1:gate′ + O2 the strands supplied to NUPACK are shown below: 1_2 gate′: 5′ UGAUGUUGUGAAGUGAGUUAAUCUAUAACCCCUUGGGGCCUCUAAACGGGUCUUGAGGGGUUUUUUG The 1_2 gate′ sequence contains the HDV ribozyme sequence. However, the HDV ribozyme structure is a pseudoknot, which NUPACK is incapable of predicting. Thus, the secondary structure of the HDV ribozyme in NUPACK does not represent its real structure. We found that the first two 5′ bases of the T7 RNAP terminator sequence n I1 (5′ CU) o were predicted to hybridize to part of the HDV ribozyme sequence on the 1_2 gate. However, this region of the ribozyme sequence is expected to be double stranded in the true ribozyme structure. To remove the influence of these spurious bases from the equilibrium analysis in NUPACK, the first C of the T7 RNAP terminator sequence was changed to an A (highlighted in yellow in the sequence above). This was done for all input sequences when analyzing these sequences in NUPACK.   Modeling was otherwise conducted as described in section V. These results provide further support of the strand displacement mechanism given that the gate presumably starts in a double stranded form in these experiments.

A. Model assumptions and reactions
RNA strand displacement reactions were modeled using ordinary differential equations derived from mass action kinetics. All modeled reactions are shown in fig. S17. In our model transcription was simplified to a first order process, whereby transcription rate is linearly proportional to the template concentration (kp*[template]). This assumption ignores transcriptional loading effects (57) S32). Because all of the transcripts in this study possess the same 5′ sequence, we assumed kp was the same for all transcripts (30). Because cotranscriptional folding is 10-fold faster than transcription (28), we assume that the gates fold instantaneously upon transcription, unless otherwise stated. Section VB discusses the rate constant values used in simulations.
Section VC describes the characterization of a leak reaction in the ctRSD system. This leak reaction was modeled by assuming that a small fraction of each ctRSD gate produced is as reactive as the designed output of the gate. Thus, a leak term was introduced in which an output is directly produced from its ctRSD gate template (kpL*[ctRSD gate template]). kpL is the first order leak transcription rate constant. For single input ctRSD gates, we found a kpL that was 3 % of kp recapitulated our experimental observations. This 3 % leak transcription was used for all single input gates. We found that a 3 % transcriptional leak for AND gates resulted in less leak than we observed in experiments. We reasoned this might be because each AND gate possesses two dsRNA domains. If we assume that each dsRNA stem has a 97 % chance of being transcribed and folded correctly, we expect the chances an AND gate is correctly produced to be (0.97) 2 = 94.1 %. Based on this analysis, we assume a 6 % transcriptional leak (kpLA) for all AND gates in the study. We also assumed the reactions between AND gates and their first inputs were irreversible because the reverse reaction is facilitated by a one base toehold. The reverse reaction between an AND gate and its final output was included in the model. Beyond the leak reaction described above, our model ignores other potential side reactions that are not expected to significantly influence dynamics. First, any gate possessing an output complementary to another gate could react via a 0 base strand displacement mechanism. This reaction was not included in the model because it occurs two to three orders of magnitude slower (11) than the designed RNA strand displacement reactions. Second, an input can react with an RNA strand displacement gate prior to ribozyme cleavage. However, a mutant ctRSD gate that could not cleave reacted much slower with input than the self-cleaving ctRSD gate (fig. S18). We assumed this side reaction would not greatly influence the observed kinetics at the low concentrations expected for the uncleaved ctRSD gate.  S19, A and B). Based on these results, the overcounting in OR gate reverse reaction rates should not influence the results for the networks we simulated in this study. A more rigorous model that tracks which gates the outputs come from could become necessary as ctRSD circuits expand.
Finally, the model does not consider any loss of T7 RNAP activity or depletion of NTPs during transcription. Thus, the model is likely to become inaccurate when simulating experimental times > (4 to 5) h, as T7 RNAP activity will have decreased significantly (58). For slow reactions that are limited by transcription (i.e. transcription of leak products), the loss of T7 RNAP activity will eventually result in a plateau in output. The model will not capture this.  In toehold mediated DNA strand displacement (DSD), the rate of the strand displacement reaction is correlated to the binding energy of the toehold. As binding energy increases with increasing toehold length, the same trend between toehold length and strand displacement rate enhancement is predicted for RSD as for DSD (35). Because rate enhancement is related to toehold binding energy, toehold sequence can also greatly influence the observed rate. For example, a strong 6 base toehold with high G-C content can result in rate constants near 10 6 L mol -1 s -1 , while weaker 6 base toehold sequences can result in rate constants closer to 10 4 L mol -1 s -1 (11). The toehold on the DNA reporter contains five A or T bases and a single G, making it a weak toehold. Thus, a rate constant of 10 4 L mol -1 s -1 was used to model the reaction between the reporter and the 1_2r strand (ksd). All reporting reactions are considered to be irreversible.
For the rate constant of the reaction between an input strand and its corresponding ctRSD gate complex (krsd), we found a value of 10 3 L mol -1 s -1 best recapitulated our experimental results. This value is at least two orders of magnitude lower than expected for a 6 base toehold with moderate GC content (11). There is some evidence that RSD reaction rate constants can be an order of magnitude lower than DSD reaction rates for short toeholds (59). Additionally, the presence of the bulky HDV ribozyme structure directly upstream of the toehold on the ctRSD gate could lower the observed reaction rate ( fig. S1). This bulky structure could sterically clash with the terminator hairpin at the 3′ end of the input strand and effectively decrease the strand displacement rate (60). In support of this hypothesis, we found the introduction of a 4 base single-stranded spacer between the HDV ribozyme motif and the toehold increased krsd to ≥10 5 L mol -1 s -1 (section VIII). We assumed the krsd value was the same for all RSD reactions in ctRSD circuits, including for both toeholds of the AND gates. For the reaction between a fuel species and an input:gate′ complex, we assumed the same reaction rate constant as between the input and the ctRSD gate.
All the RNA strand displacement reactions in this study are reversible ( fig. S3). Estimation of the reverse reaction rate constants based on toehold length and sequence alone is confounded by reverse reactions replacing a G-C pair with a G-U wobble in the first two to three bases of branch migration ( fig. S3). This will reduce the rate of strand displacement, but the amount of this reduction is unknown. In DNA strand displacement, introduction of a mismatch at a similar position during branch migration can reduce the reaction rate by two to three orders of magnitude; presumably a G-U wobble would have a slightly less pronounced effect. Because we did not find an estimate for a comparable system in the literature, we estimated the reverse reaction rate constants from an equilibrium analysis of the strand displacement products. We used NUPACK 3.2.2 (42) to calculate the equilibrium constant (Keq) for each complementary gate and input sequence (section III). The reverse reaction rate constant (krev) was determined from the equilibrium constant as krev = krsd/Keq. Based on this analysis, we found gates with outputs possessing the b-toehold have reverse rate constants nearly three orders of magnitude lower than the forward rate constants. Gates with outputs possessing the a-toehold have reverse reaction rate constants only one order of magnitude lower than the forward rates (table S2). To simplify the model, we assumed a single reverse rate constant for gates with b-toeholds (5 L mol -1 s -1 ) and for gates with a-toeholds (270 L mol -1 s -1 ). The reaction rate constants must be 10-fold larger than these values to begin to influence model predictions ( fig. S19).
The HDV ribozyme cleavage rate constant was estimated as 0.25 min -1 ( fig. S12), consistent with previously reported in vitro values (54).
All the rate constants used in simulations are presented in table S3.

% of kp
Calibrated to data k pLA (s -1 )

% of kp
Calibrated to data

C. Modeling and characterizing leak
In our experiments, we observed a leak in which transcription of the 1_2r gate template in the absence of the I1 template resulted in a slow increase in DNA reporter signal. This leak reaction increased with increasing concentrations of T7 RNAP, i.e., the leak increased with increasing transcription rate (fig. S20). The initial model of ctRSD circuits did not include terms capable of producing this leak ( fig. S21, A and B). To include this observed leak in the model, we investigated three potential models of leak the pathway: Models 1, 2, and 3 in fig. S21C. Both Model 2 and Model 3 can recapitulate the experimental data, but only Model 3 is consistent with the experimental results presented in fig. S22. Unless otherwise stated, all other simulations included the leak reactions depicted in Model 3.
We ruled out two additional models for leak based on previous data and literature: 1) Short transcripts produced during abortive cycling by T7 RNAP could include part of the output domains and react with the DNA reporter. This model was considered unlikely because short abortive transcripts typically range from (2 to 12) nucleotides (32) but the gate transcripts possess a 17 nucleotide hairpin sequence at their 5' end. Thus, any short transcripts produced during abortive cycling should not contain sequence complementarity with the reporter.
2) The ribozyme rapidly cleaves during transcription and releases the output before the bottom strand of the gate (gate') is produced. The output strand could then irreversibly react with the DNA reporter before hybridizing to form a dsRNA gate. This model was considered unlikely because we measured the HDV ribozyme cleavage rate constant to bẽ 0.25 per min (~0.004 per s) in our assay conditions ( fig. S12). This value is consistent with previously published values for HDV ribozyme self-cleavage in vitro (54). From our simulation results, the transcription rate constant in our experiments was ~ 0.01 per s, indicating transcription proceeds much faster than ribozyme cleavage. The results in fig.  S22 are also inconsistent with this model of leak because the leak is still observed even when the RNA gate is transcribed in isolation. . From left to right, the concentration of T7 RNAP is increased. Increasing the polymerase concentration increased the rate of output production both with and without input. The transcriptional load in the sample with 50 nmol/L I1 and 25 nmol/L 1_2r gate is higher than the sample with 25 nmol/L 1_2r gate alone. To ensure the transcriptional load between the samples with and without I1 is the same, we tested another sample with the Io template (which produces an input RNA that does not react with the 1_2r gate) instead of the I1 template. Inclusion of the Io template reduces the leak from the 1_2r gate and ensures the same transcriptional load across samples for comparison. 500 nmol/L of DNA reporter was used in all samples. (C) Schematics of the different leak models investigated. Each model includes a different pathway for a leak reaction between the 1_2r gate and the DNA reporter (red reaction lines). In Model 1, the leak is modeled as a 0 base toehold reaction between the cleaved 1_2r gate and the DNA reporter. In Model 2, an additional folding step for the 1_2r gate transcript was introduced into the model. This model assumes the 1_2r gate RNA can react with the DNA reporter complex before the gate has folded. Because such a reaction would use the 6 base b-toehold-as does the designed reaction between the 1_2r strand-the leak was assumed to have the same rate constant as the designed reporting reaction (ksd). In Model 3, it is assumed that a certain percentage of the 1_2r gate transcripts produced are truncated or misfolded. In the case of a truncated product, the b-toehold would be exposed, and thus the truncated product could react with the DNA reporter complex with the same rate constant as the designed reporting reaction.
(D) Simulation results (dashed lines) for Model 1 compared to experimental results (solid lines). In the simulations, a kleak of 15 L mol -1 s-1 was used for the 0 base toehold reaction between the 1_2r gate complex and the DNA reporter. This is an order of magnitude higher than reported previously (11). (E) Simulation results (dashed lines) for Model 2 compared to experimental results (solid lines). In the simulations, a kfold of 0.15 s -1 was used. Considering that cotranscriptional folding occurs much faster than transcription (28), the kfold parameter may be taken as the time required to produce the transcript, during which the nascent transcript could react with the DNA reporter. A kfold of 0.15 s -1 corresponds to a transcript produced every 6.67 s, and this corresponds to the transcription rate of ≈27 nt/s for the 183 nt 1_2r gate transcript. This transcription rate is within a factor of 1.5 of previously reported transcription rates for T7 RNAP (61), supporting the feasibility of the kfold parameter that recapitulates the experimental data. (F) Simulation results (dashed lines) for Model 3 compared to experimental results (solid lines). In the simulations, a production rate of truncated 1_2r gate products (kp,L) that was 3 % of the production rate of correct products (kp) was used. The reaction between the DNA reporter and the truncated 1_2r gate product was assumed to have the same rate constant (ksd) as the reaction between the DNA reporter and the 1_2r strand. All other rate constants are in table S3. The experimental results are also presented in Figure 2 of the main text.  Fig. S23: Denaturing gel of ctRSD gates with 1, 3, 4, and 5 input domains. All gates self-cleave as designed. Transcription and gel electrophoresis were conducted as described in the Methods of the main text.

Fig. S24: Native gels of 3&1_2 AND gates with different sized a′ internal loops between input domains 3 and 1.
(A) Schematics of the desired AND gate design. The 3&1_2 gate is designed so that I3 must first react with the gate to expose the toehold for I1 to react and release the output. To accomplish this, we introduced an internal loop in the a:a′-domain between the 3 and 1 duplexes on the AND gate. The gray domains in the internal loops are short linker domains added to reduce strain between the 3 and 1 duplexes. A tradeoff to consider when introducing these internal loops pertains to the following: the a′-loop is the toehold for I1. The longer the a′-loop, the higher the rate at which I1 can react with the AND gate in the absence of I3. We designed AND gates with (3, 4, 5, and 6) base a′-loops to find a design that favored the reaction with I3, while disfavoring the reaction with I1 alone. (B) Schematics of the different AND gate designs tested. (C) Native gel shift assay results for the AND gate variants in (B). In these experiments, a higher molecular weight product should appear both when I3 is present and when I1 and I3 are present. Additionally, I1 by itself should not produce a higher molecular weight product. The AND gate with a 3 base a′-loop does not react with I3 or I1+I3. The AND gate with a 4 base a′-loop reacts ≈50 % with I3 and I1+I3. The AND gates with 5 base or 6 base a′-loops fully react with I3 and I1+I3. The larger the a′-loop, the more likely I1 will react with the AND gate in the absence of input. Thus, we selected the 5 base a′-loop design. The samples were run on the gels 2 h after the DNA templates were degraded with DNase I. Experiments were otherwise conducted as described in the Methods of the main text. The three images were taken from three different gels. Fig. S25: Characterization of AND gate reactions. AND gates were analyzed with the native RNA gel shift assay (A) and the DNA reporter assay (B). The 5&4_2r experiment was conducted using the same gate, input(s), and T7 RNAP concentrations as the 3&1_2r circuit element (table S4). For the gel electrophoresis results, gate and input templates were at 25 nmol/L and 50 nmol/L (2x), respectively. Electrophoresis was conducted 1 h after DNase I addition. The gate′ strand is from the 3&1_2r gate. The first seven lanes of the gel are also presented in Figure 4E of the main text. The DNA reporter 3&1_2r results are also presented in Figure 4F of the main text.
Across our experiments there were two minor deviations from simulation predictions. Deviation 1: There was lower leak than predicted between ctRSD gates, which could be the result of steric hindrance between leak products and gates ( fig. S26). Deviation 2: The gates that take I4 as an input reacted slower than the gates that take other inputs. The 4_2r gate was noticably slower than the three other single input gates tested ( Figure 3A of the main text), and the 4_1 gate was slower than the 5_1 gate in a two-layer cascade (fig. S27B). The 4_1 gate was also slower in a logic cascade than the 5_1 gate ( fig. S27D). We hypothesized that the strand displacement rate constant for gates that take I4 as an input could be lower than the other domains due to the high UA content at the start of branch migration ( fig. S3). Similar sequences have been shown to significantly decrease overall strand displacement kinetics (44). Decreasing krsd 4-fold in our simulations for just the gates that take I4 as an input resulted in better model agreement across experiments ( fig. S27). Thus, the leak product likely reacts with the DNA reporter at a similar rate as a single-stranded output. (B and C) Single-input ctRSD gates (B) and ctRSD AND gates (C) have duplexes upstream of their 5′ toehold, which could sterically hinder a reaction with a bulky leak product. Note the leak products shown here are hypothetical but are representative of the true leak products, which are likely bulkier than their ssRNA output strands. (D) Schematics of the single-stranded 5_1 output strand and the 5_1 leak hairpin product used in the experiments in (E) and (F). The 5_1 HDV cau transcript contained the output strand of the 5_1 gate, followed by the HDV ribozyme and a 22 base sequence composed of only C, A, U bases. Upon transcription, this transcript cleaves to produce the output from the 5_1 gate. The 5_1 leak hairpin transcript was truncated such that the a-toehold of the 5_1 gate output was exposed. Further, an inactive ribozyme variant (xRz) was used to ensure the transcript remained in a hairpin structure. (E) Experimental (solid) and simulated (dashed) reporter signal during cotranscription of the 1_2r gate and the RNA products in (D). [T7 RNAP] = 1 U/µL. For the simulations, a kp of 0.01 s -1 was used, and the RNA strand displacement rate constant between the 5_1 leak product and the gates was varied to recapitulate the experimental kinetics. The reaction rate constant with the leak reaction was approximately two orders of magnitude lower than the rate constant for the reaction with the single-stranded 5_1 output. The rate constants are given inside the plots.  Figure 3A of the main text with simulation results using krsd = 1x10 3 L mol -1 s -1 (left) or 2.5x10 2 L mol -1 s -1 (right). (B) A two-layer cascade of I4 to 4_1 to 1_2r (purple) or I5 to 5_1 to 1_2r (teal) with simulation results using krsd = 1x10 3 L mol -1 s -1 (left) or 2.5x10 2 L mol -1 s -1 (right). (C) The four-layer cascade from Figure 5B of the main text with simulation results using krsd = 1x10 3 L mol -1 s -1 (left) or 2.5x10 2 L mol -1 s -1 (right). (D) The logic cascade containing the 4_1 gate from Figure 5E with simulation results using krsd = 1x10 3 L mol -1 s -1 (left) or 2.5x10 2 L mol -1 s -1 (right). All other rate constants are presented in table S3.

VIII. 1_2r gates with different toehold lengths
The kinetics of toehold-mediated strand displacement reactions can be controlled by toehold length. Here, we explore how toehold length influenced the kinetics of ctRSD circuit reactions. The initial design for the 1_2r gate included a 6 base single-stranded input toehold, which we would expect to result in a rate constant near the maximum theoretical limit (10 6 L mol -1 s -1 ) (11,36). However, our simulations indicated that the forward strand displacement rate constant between the 1_2r gate and I1 was only 10 3 L mol -1 s -1 . We theorized steric hindrance between the ribozyme and the input strand could result in slower strand displacement because the 6 base toehold is directly adjacent to the bulky HDV ribozyme motif ( fig. S1). Thus, in addition exploring the influence of toehold length on kinetics, we also explored the influence of including a single-stranded spacer sequence between the ribozyme motif and the toehold. To do this, we designed 1_2r gates with (6, 8, 10, and 12) base toeholds and I1 variants possessing (4, 6, 8, or 10) base toeholds and combinatorially tested all gate and input combinations. Schematics with sequences are presented in fig. S6. We first confirmed increasing toehold length did not influence gate folding and/or cleavage. Fig. S28 demonstrates that the 1_2r gate toehold variants cleaved as designed. Fig. S29 shows that increasing toehold length did not increase leak with the DNA reporter, suggesting proper folding.
We next evaluated RSD kinetics for all gate and input toehold length combinations. These experiments encompassed toehold lengths of 4 bases to 10 bases with spacer lengths varying from (0 to 8) bases depending on the input toehold length (fig. S30, A and B). In these experiments, we were not able to resolve reaction rate constants greater than 10 5 L mol -1 s -1 ( fig.  S30C). When the strand displacement reaction rate gets this high, the overall rate of output release becomes limited by transcription and gate cleavage, rather than strand displacement. Thus, we report all reaction rate constants near this 10 5 L mol -1 s -1 limit as ≥10 5 L mol -1 s -1 ( fig.  S30B). Fig. S30D show the kinetic traces for each gate and input toehold combination, highlighting the influence of spacer length on reaction kinetics for each input toehold length. Inclusion of a spacer generally increases reaction rate, but the spacer length that saturates the reaction rate decreases as input toehold length increases. An explanation for this observation could be: the weaker the input binding energy, the greater the influence of steric hindrance on the reaction rate. For example, the input with a 4 base toehold binds weakly to the 1_2r gate toehold, so a long spacer is required to completely remove any effect of steric hindrance. Conversely, for the input with a 10 base toehold, the same kinetics are observed for a 0 base and 2 base spacer. In this case, the input with the 10 base toehold can be viewed as an input with a 6 base toehold binding to a gate with a 4 base spacer, or an input with an 8 base toehold binding to a gate with a 2 base spacer. Both those reactions occur at a rate near the maximum value. Put another way, once the input toehold is long enough, increasing the length of the a-toehold (input) and a′toehold (gate) together has almost the same effect as simply increasing the spacer length, i.e. increasing the a′-toehold (gate) without increasing the a-toehold (input). In support of this hypothesis, the 8 base a-toehold (input) and 8 base a′-toehold (gate) reaction rate constant is close to the 6 base a-toehold (input) and 8 base a′-toehold (gate) reaction rate constant ( fig.  S30B). Fig. S30E shows the kinetic traces for each gate and input toehold combination, highlighting the influence of toehold length on reaction kinetics for each spacer length. With the exception of the input with a 4 base a-toehold, most of the changes in kinetics observed across toehold length can be attributed to the increase in a′-toehold (spacer) length. For example, with long enough spacers, inputs with (6, 8, and 10) base toeholds exhibit strand displacement constants close to the maximum value (≥10 5 L mol -1 s -1 ).
How do these results compare to previous studies of toehold-mediated strand displacement kinetics? For traditional DNA (11) and RNA strand displacement (36), in which double-stranded complexes are pre-annealed and gate toeholds have no secondary structure upstream, toeholds ≥6 bases should result in reaction rate constants at the theoretical maximum of 10 6 L mol -1 s -1 . We found similar results for ctRSD circuits when using a long enough spacer between the HDV ribozyme and the toehold. Regarding the input with a 4 base a-toehold, the reaction between this input and any of the 1_2r gates has a much lower thermodynamic driving force than the other input toeholds tested. This is because the 1_2r gates all possess a 6 base reverse toehold, i.e. completion of the forward strand displacement reaction results in a net loss of two base pairs compared to the intact 1_2r gate. The rate constant for a DNA strand displacement reaction between an input with a 4 base toehold and a gate with a 6 base reverse toehold (b-toehold) was measured to be between (10 2 and 10 3 ) L mol -1 s -1 . This aligns with our estimated rate constant of 2x10 2 L mol -1 s -1 for between the 4 base input toehold variant and ctRSD gates with either a 6 base or 8 base spacers ( fig. S30B). Together, these results suggest that ctRSD circuits should possess the same kinetic control of traditional toehold-mediated strand displacement, provided appropriate spacers are used.
Steric hindrance introduced by the ribozyme could also be used as an additional feature to tune strand displacement rates. Changing the spacer length adjacent to the ribozyme allows different strand displacement rate constants to be obtained, without needing to change the input's toehold length. For the 6 base a-toehold, varying spacer length changed the strand displacement rate constant by two orders of magnitude. Further experiments are necessary to fully characterize this design space. Fig. S28: Denaturing gel of 1_2r gates with (6, 8, 10, and 12) base a′-toeholds. All gates self-cleave as designed. Transcription and gel electrophoresis were conducted as described in the Methods of the main text.

IX. Potential advantages of ctRSD circuits compared to DNA-based circuits
Should ctRSD circuits continue to prove as predictable and programmable as DNA-based circuits, ctRSD could serve as a more versatile alternative to DNA computing. Such a shift could be justified given the high fidelity and decreasing price of gene synthesis. Integrated DNA Technologies currently reports ≈80 % of 30 base DNA oligonucleotides are the correct product compared to ≈100 % for gBlocks of >125 bases (62). The low fidelity of DNA oligonucleotide synthesis requires the strands to be purified with gel electrophoresis and many DNA computing papers report the purification of individual dsDNA circuit complexes to obtain desired circuit function (4,7). For ctRSD circuits the high-fidelity gBlock synthesis is followed by a highfidelity PCR step (<0.25 % error (63)) and high-fidelity transcription-T7 RNAP's nucleotide substitution rate is less than 1 in 17,000 bases (64). Further, encoding the dsRNA complex as a single transcript ensures the proper stoichiometry between the two gate strands, reducing leak pathways (29). Thus, ctRSD circuits remove the need for purification of circuit components before operation, greatly simplifying the workflow. Further, the per nanomole cost of a ctRSD gate template can be reduced to nearly that of analogous DNA gates with a few modifications to the protocol presented here (section IXA).
Another advantage of using transcriptionally encoded circuits over DNA strand displacement circuits is the long-term stability of long linear DNA templates and DNA plasmids. For example, in many biosensor (26) and diagnostic (65) applications, circuit components are freeze dried for long-term storage and ease of transportation. These freeze-dried circuits are then activated by adding a liquid sample at the point of need. Both linear DNA templates on the order of 300 bases and DNA plasmids have been shown to remain stable for months after freeze drying. Short DNA strand displacement duplexes show significant decrease in performance only one week after freeze drying (27).

A. Cost analysis
Cost of preparing DNA templates for ctRSD circuits: In this study, the gBlocks were price fixed from (125 to 500) bases and were $89 USD each for 250 ng. The gBlocks were resuspended in 25 µL of Buffer EB. For PCR, 1.5 µL of gBlock DNA was used in a 75 µL reaction and typically yielded 50 µL of 600 nmol/L product after purification. This is 30 pmol of DNA template per reaction, and 16.67 reactions can be conducted using the 25 µL of gBlock DNA, resulting in 0.5 nmol of total DNA template for a given gBlock order. This comes to $178 per nmol of template. The Phusion Master-Mix used for gBlock PCRs cost $782 for 500 reactions, which is $1.56 per reaction and $26 to produce all 0.5 nmol of DNA. This results in an additional $52 per nmol of DNA. Finally, the cost of the DNA primers used in the PCR must be added. These 25 base oligos cost $9.25 for 180 µg and 0.5 µmol/L (0.31 µg) were used in each reaction, which is $0.016 per reaction and $0.26 to produce the total 0.5 nmol of product ($0.52 per nmol). In total, the DNA templates for ctRSD circuits cost: $178 per nmol for the gBlock DNA, $52 per nmol of DNA for the Phusion Master-Mix, and $0.52 per nmol for the DNA primers = $230.52 per nmol.
Note that the main cost is the initial purchase of the gBlock DNA. This cost could be significantly reduced by purchasing the DNA in bulk as eBlocks from IDT. eBlocks require an order of at least 24 sequences but cost only $0.06 per base, which is $18 for a 300 base template (the minimum length for ordering). This would drop the cost to $88.52 per nmol of DNA template. Further, a gBlock or eBlock template only needs to be ordered a single time from IDT, and once the DNA from that order has been exhausted, the PCR products of the gBlock DNA can themselves be PCR amplified to produce more template. A single gBlock PCR could be used to conduct 250 more PCR experiments, which yields 7.5 nmol of DNA template. If PCR of a PCR amplified gBlock is conducted just once, the cost for a DNA template is dropped to $11.87 per nmol for gBlocks and $2.4 per nmol for eBlocks. This amounts to $64.39 and $54.94, respectively. Here the primary cost is from PCR reagents.
Cost of preparing a DNA gates for DNA strand displacement: For DNA oligos, IDT charges $0.70 for oligos at the 100 nmol scale, which is required for purification. If these oligos are ordered with polyacrylamide gel electrophoresis (PAGE) purification, the added cost is $60. PAGE purification has a guaranteed yield of 2 nmol of final product.
A PAGE purified 26 base oligo that serves as the output strand of a DNA gate thus costs $78.2 for a guaranteed 2 nmol of DNA or $39.10 per nmol. A 22 base gate′ strand is $75.40 for 2 nmol ($37.7 per nmol). Thus, at the guaranteed yield, the cost is $76.80 per nmol of gate.  . S31). To ensure the same transcription load across different samples for the same ctRSD element or circuit, a template producing an unreactive input (Io) was added so that all samples had the same total template concentration. The total concentration of templates for each experiment is also presented in table S4. Because the transcription rate differed across many experiments, the first order rate constant (kp) used to model transcription had to be calibrated for a given T7 RNAP and total template concentration ( fig. S32). The first order rate constant (kp) calibrated for each experiment is presented in table S4. This transcription rate calibration should also calibrate for batch to batch variation in T7 RNAP stocks. Other than the differences arising from different experimental conditions or T7 RNAP batches, replicate measurements of circuit kinetics varied between 2 % to 5 % (section XC).   The 1_2r HDV cau template, which constitutively produces the O2r strand, was used as a reference sample for each experiment. The Io template was added to bring the total template concentration in the reference sample to the total concentration used in the experimental samples. The reference sample was then used to calibrate the first order transcription rate constant, kp, for simulation of the experimental samples. Fig. S33: Analysis of variation in ctRSD circuit experiments. Experiments represent independently prepared technical replicates for ctRSD circuit reactions using the 1_2r gate (A) or the 5&4_1+3&1_2r gates (B). Left plots show data for three independently prepared replicates of the same ctRSD circuit reaction. The replicates were prepared from the same stock solutions on the same day. The right plots show the mean of the three replicates; error bars represent one standard deviation. The standard deviation was < 1.5 % from the mean value at each time point for (A) and < 5 % from the mean value at each time point for (B). In (A), reactions included 25 nmol/L of the 1_2r gate template, 12.5 nmol/L of the I1 template, and Io template to bring the total template to 50 nmol/L in both reactions. In (B), reactions included 25 nmol/L of the 5&4_1, 25 nmol/L of the 3&1_2r gate, and 25 nmol/L of the input templates. For the samples containing I3 only, 50 nmol/L of the Io template was added to bring the total template concentration to 75 nmol/L. In all experiments, 500 nmol/L of the DNA reporter and 2 U/µL of T7 RNAP was used. Left plots show data for three independent replicates of the same ctRSD circuit reaction. Reactions included 25 nmol/L of the 1_2r gate template, 500 nmol/L of the DNA reporter, 1 U/µL of T7 RNAP, and 50 nmol/L of the I1 or Io templates. The three individual replicates were prepared independently and tested on separate days, with the second and third replicates conducted 5 and 19 days after the first replicate, respectively. Dark colored lines represent the oldest replicate; data also presented in fig. S20B, middle panel. Medium colored lines represent the second oldest replicate; data also presented in Figure 2D, 2x I1, of the main text. Light colored lines represent the newest replicate; data also presented in fig. S18B. The right plot shows the mean of the three replicates; error bars represent one standard deviation. The standard deviation is < 3 % from the mean value at each time point.